Variational quantum algorithm for generalized eigenvalue problems and its application to the finite element method

Abstract

Generalized eigenvalue problems (GEPs) play an important role in the variety of fields including engineering, machine learning and quantum chemistry. Especially, many problems in these fields can be reduced to finding the minimum or maximum eigenvalue of GEPs. One of the key problems to handle GEPs is that the memory usage and computational complexity explode as the size of the system of interest grows. This paper aims at extending sequential quantum optimizers for GEPs. Sequential quantum optimizers are a family of algorithms that iteratively solve the analytical optimization of single-qubit gates in a coordinate descent manner. The contribution of this paper is as follows. First, we formulate the GEP as the minimization/maximization problem of the fractional form of the expectations of two Hermitians. We then showed that the fractional objective function can be analytically minimized or maximized with respect to a single-qubit gate by solving a GEP of a 4 $\times$ 4 matrix. Second, we show that a system of linear equations (SLE) characterized by a positive-definite Hermitian can be formulated as a GEP and thus be attacked using the proposed method. Finally, we demonstrate two applications to important engineering problems formulated with the finite element method. Through the demonstration, we have the following bonus finding; a problem having a real-valued solution can be solved more effectively using quantum gates generating a complex-valued state vector, which demonstrates the effectiveness of the proposed method.

19 pages, 10 figures

References in corpus (5)

• Quantum algorithm for solving linear systems of equations
• Simulated Quantum Computation of Molecular Energies
• Variational Quantum Solutions to the Advection-Diffusion Equation for Applications in Fluid Dynamics
• Application of a variational hybrid quantum-classical algorithm to heat conduction equation
• Excited-state molecular dynamics simulation based on variational quantum algorithms

Cited by in corpus (3)

• Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits
• Optimal Parameter Configurations for Sequential Optimization of Variational Quantum Eigensolver
• Quantum Realization of the Finite Element Method